Indeed, the measure \(\mu \)  can be constructed via Caratheodory's extension theorem applied to the
measure \(\hat{\mu }\)  on the semiring
\({S}\lbrace {P}_{\infty }(\pi ) : \pi \in {P}_{n}\setminus \mathbf {0}_n, \hspace{6.0pt} n\in \mathbb {N}\rbrace \)  defined analogously
through (REF ).
The finite-additivity of \(\hat{\mu }\)  follows easily from (REF ). Infinite-additivity
follows from the fact that \({P}_{\infty }\)  is compact (see Lemma 2.6 in [1]})
and the fact that the elements of \({S}\)  are both open and closed. Indeed, the latter implies that if \({S}\ni A = \cup _{i=1}^\infty A_i\) , \(A_i \in {S}\) , then, in fact,
there exists a finite collection of indices \(i_1,\dots ,i_k\)  such that \(A=\cup _{j=1}^k A_{i_j}\) . Thus, infinite-additivity follows from finite-additivity. The measure \(\mu \)  being
invariant under finite permutations (by construction), Theorem 4.2 in [1]} ensures the existence
of a measure \(\Xi \)  on \({P}_{[0,1]}\)  such that
\(\mu (\pi )={Q}_{\Xi ,n}(\pi ),\quad \forall n\in \mathbb {N}, \pi \in {P}_{n}\setminus \mathbf {0}_n.\) 
